Affine Cone Research Articles

Counting gauge invariant operators in SQCD with classical gauge groups

We use the plethystic programme and the Molien-Weyl fomula to compute generating functions, or Hilbert Series, which count gauge invariant operators in SQCD with the SO and Sp gauge groups. The character expansion technique indicates how the global symmetries are encoded in the generating functions. We obtain the full character expansion for each theory with arbitrary numbers of colours and flavours. We study the orientifold action on SQCD with the SU gauge group and examine how it gives rise to SQCD with the SO and Sp gauge groups. We establish that the classical moduli space of SQCD is not only irreducible, but is also an affine Calabi-Yau cone over a weighted projective variety.

Typicality, black hole microstates and superconformal field theories

We analyze the structure of heavy multitrace BPS operators in N = 1 superconformal quiver gauge theories that arise on the worldvolume of D3-branes on an affine toric cone. We exhibit a geometric procedure for counting heavy mesonic operators with given U(1) charges. We show that for any fixed linear combination of the U(1) charges, the entropy is maximized when the charges are in certain ratios. This selects preferred directions in the charge space that can be determined with the help of a piece of string. We show that almost all heavy mesonic operators of fixed U(1) charges share a universal structure. This universality reflects the properties of the dual extremal black holes whose microstates they create. We also interpret our results in terms of typical configurations of dual giant gravitons in AdS space.

A multisplitting method for symmetrical affine second-order cone complementarity problem

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. In this paper, a multisplitting method for the symmetrical affine SOCCP is developed and analysed. This method makes use of a set of different splittings of the coefficient matrix. Convergence is established for the proposed method. Moreover, an MAOR-like method is considered to solve the subproblems. We also report some simple numerical results for the proposed method.

Log-terminal singularities and vanishing theorems via non-standard tight closure

Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over C \mathbb C , in terms of purity properties of ultraproducts of characteristic p p Frobenii. As a first application we obtain a Boutot-type theorem for log-terminal singularities: given a pure morphism Y → X Y\to X between affine Q \mathbb Q -Gorenstein varieties of finite type over C \mathbb C , if Y Y has at most log-terminal singularities, then so does X X . The second application is the Vanishing for Maps of Tor for log-terminal singularities: if A ⊆ R A\subseteq R is a Noether Normalization of a finitely generated C \mathbb C -algebra R R and S S is an R R -algebra of finite type with log-terminal singularities, then the natural morphism Tor i A ⁡ ( M , R ) → Tor i A ⁡ ( M , S ) \operatorname {Tor}^A_i(M,R) \to \operatorname {Tor}^A_i(M,S) is zero, for every A A -module M M and every i ≥ 1 i\geq 1 . The final application is Kawamata-Viehweg Vanishing for a connected projective variety X X of finite type over C \mathbb C whose affine cone has a log-terminal vertex (for some choice of polarization). As a corollary, we obtain a proof of the following conjecture of Smith: if G G is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety X X , then for any numerically effective line bundle L \mathcal L on any GIT quotient Y := X / / G Y:=X/\!/G , each cohomology module H i ( Y , L ) H^i(Y,\mathcal L) vanishes for i > 0 i>0 , and, if L \mathcal L is moreover big, then H i ( Y , L − 1 ) H^i(Y,\mathcal L^{-1}) vanishes for i > dim ⁡ Y i>\operatorname {dim}Y .

A matrix-splitting method for symmetric affine second-order cone complementarity problems

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. The purpose of this paper is to propose an iterative method for the symmetric affine SOCCP that is based on the idea of matrix splitting. Matrix-splitting methods have originally been developed for the solution of the system of linear equations and have subsequently been extended to the LCP and the affine variational inequality problem. In this paper, we first give conditions under which the matrix-splitting method converges to a solution of the affine SOCCP. We then present, as a particular realization of the matrix-splitting method, the block successive overrelaxation (SOR) method for the affine SOCCP involving a positive definite matrix, and propose an efficient method for solving subproblems. Finally, we report some numerical results with the proposed algorithm, where promising results are obtained especially for problems with sparse matrices.

Equivariant Embeddings into Smooth Toric Varieties

AbstractWe characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well-known embedding theorem of Sumihiro on quasiprojectiveG-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projectiveG-variety to a more general framework.

The Volume of some Non-spherical Horizons and the AdS/CFT Correspondence

We calculate the volumes of a large class of Einstein manifolds, namely Sasaki-Einstein manifolds which are the bases of Ricci-flat affine cones described by polynomial embedding relations in C^n. These volumes are important because they allow us to extend and test the AdS/CFT correspondence. We use these volumes to extend the central charge calculation of Gubser (1998) to the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999). These volumes also allow one to quantize precisely the D-brane flux of the AdS supergravity solution. We end by demonstrating a relationship between the volumes of these Einstein spaces and the number of holomorphic polynomials (which correspond to chiral primary operators in the field theory dual) on the corresponding affine cone.

Todd Classes of Affine Cones of Grassmannians

A ring A which is a homomorphic image of a regular local ring S is said to be a Roberts ring if τA/S([A]) = [Spec A]dim A, where τA/S is the Riemann-Roch map for Spec A. Such rings satisfy a vanishing theorem for intersection multiplicities, as was proved by P. Roberts. It is known that complete intersections are Roberts rings, and the first author showed that a determinantal ring is a Roberts ring precisely if it is a complete intersection. Let Ad(n) denote the affine cone of the Grassmann variety Gd(n) under the Plucker embedding. In this paper, we determine precisely when Ad(n) is a Roberts ring.

The Brauer group of an affine cone

Let V be a reduced projective hypersurface in Pv and C the affine cone over V in X=Av+1. We employ cohomological methods to study the Brauer groups of the varieties C, C-P, X-C and X¯-C¯, where P is the vertex of the cone, X¯=Pv+1 and C̄ is the completion of C in X̄.

On Complex Homogeneous Spaces with Top Homology in Codimension Two

AbstractDefine dx to be the codimension of the top nonvanishing homology group of the manifold X with coefficients in 2. We investigate homogeneous spaces X := G/H, where G is a connected complex Lie group and H is a closed complex subgroup for which dx = 1,2 and O(X) ≠ ℂ. There exists a fibration π: G/H → G/U such that G/U is holomorphically separable and π*(O(G/U)) = O(G/H), see [11]. We prove the following. If dx = 1, then F := U/H is compact and connected and Y :=G/U is an affine cone with its vertex removed. If dx = 2, then either F is connected with dF = 1 and Y is an affine cone with its vertex removed, or F is compact and connected and dy = 2, where Y is ℂ, the affine quadric Q2, ℙ2 — Q (with Q a quadric curve) or a homogeneous holomorphic * -bundle over an affine cone minus its vertex which is itself an algebraic principal bundle or which admits a two-to-one covering that is.

Picard numbers of surfaces in 3-dimensional weighted projective spaces

Let q0 . . . . . q3 be positive integers. Consider the weighted complex projective 3-space F = F(qo, q~, q2, q3) = Proj(r x l , x2, x3]) where xl has weight ql. We will assume that gcd{q~,qa, qr} = 1 whenever # {~,fl, 7} = 3 (this is not a restriction in fact, see 1-3] w For d ~ N we let ~'~d be the family of surfaces of degree d in IP, i.e. the schemes of zeros of sections in (~F(d). Such a section is called quasi-smooth if its affine cone in ~4 has an isolated singular point at the origin. If X e ~d is quasi-smooth, then X is a V-manifold ([5], Lemma 1 p. 216). Hence H2(X, Q) has a pure Hodge structure and we can ask

On projective Cohen-Macaulayness of a Del Pezzo surface embedded by a complete linear system

Let k be an algebraically closed field. We understand by a Del Pezzo surface X over k a non-singular rational surface on which the anti-canonical sheaf ―wx is ample. We call the self-intersectionnumber d=a)x of wx the degree of X, then we get that 1^J^E9. It is well known that X is isomorphic to PlxP which has degree 8, or an image of P2 under a monoidal transformation with center the union of r―9―d points which satisfiesthe following conditions: (a) no three of them lie on a line; (b) no six of them lie on a conic; (c) there are no cubics which pass through seven of them and have a double point at the eighth point. Conversely any surface described above is a Del Pezzo surface of the corresponding degree ([8,in, Theorem 1]). It is also well known that ―o)x is very ample when d^3 and that ample divisors on X of degree 3, which is a cubic surface, are very ample too. In this paper we will get that ample divisorson X of degree d^3 are very ample and that ample divisors on X of degree 2 [resp. 1] other than ―o>x [resp. ―o)x nor ―2^x] ore very ample. A closed subscheme V in PN is said to be projectively Cohen-Macaulay if its affine cone is Cohen-Macaulay. It is equivalent to that H1(PN,Jv(m))=0 for every meZ and H\V, Ov(m))-0 for every meZ and 0<?<dim V. In this paper, we will get that (p\o\(X) is projectively Cohen-Macaulay for a very ample divisor D on X, where <p[D[ is the morphism from X to JPdiml£ defined by the complete linear system ID I of D. We also study the homogeneous ideal I(D)=Ker \sr(D) ― c I\nD) defining <pim(X). These results will be stated and proved in §3 and §5. The fourth section will be devoted to a study on ―ncox of a Del Pezzo surface X of degree 1 or 2. In §1 we will compute the dimension h\D) of the z-th cohomology group H\X,Ox{D)) of the invertible sheaf OxiP) corresponding to a divisor D.

Algebraisation of Some Formal Vector Bundles

In this paper I consider the question of when formal vector bundles on the formal scheme obtained by completing the punctured spectrum of a ring of formal power series along a closed subscheme are algebraic. This leads to a generalization of Zariski's theorem about purity of branch loci to rings whose embedding dimension is not much bigger than their Krull dimension. By applying it to the local ring in the vertex of the affine cone over some projective space we see that for an irreducible scheme X _ Pk, where k is an algebraically closed field, the algebraic fundamental group vanishes, if dim(X) > n/2. This result was also obtained by Fulton and Hansen (see [FHI), and special cases were obtained previously by Barth (where k = C, and X is